Where there are multiple forces acting on the same object, draw a free-body diag

Question

Where there are multiple forces acting on the same object, draw a free-body diagram (i.e a diagram where each vector for each of the different forces is shown): A book on a table has a coefficient of static friction of 0.1 and a coefficient of kinetic friction of 0.05. The book has a mass of 1kg. The table has a length of 3m. Suppose you gently tilt the table until the book starts to slip and then leave the angle of the book at that critical angle afterwards. How long will it take the book to slide to the end of the table?

Explanation / Answer

Given ,
Static Friction us = 0.1
Kinetic Friction uk = 0.05
Let us assume table is tilted at an angle .

Weight of Block acting along the incline = mg * sin()
Normal Force = mg * cos()

To begin movement , Frictional force = Weight of book acting along the incline
Frictional Force at rest = Static Friction * Normal Force
Frictionaal Force at rest = us * mg * cos()
us * mg * cos() = mg * sin()
0.1 = tan()
= 5.70

Now, The table is tiled at the same angle, Weight will be same along the inclline but frictional force would change to kinetic Friction which would cause acceleration-
ma = mg * sin() -  uk * mg * cos()
a = 9.8 * (sin(5.7) - 0.05 * cos(5.7))
a = 0.485 m/s^2

Length of Table = 3m
Initial Velocity = 0
Acceleration a = 0.485 m/s^2
S = u*t + 0.5 at^2
3 = 0 + 0.5 * 0.485 *t^2
t = sqrt(6/0.485) s
t = 3.52 s
Time it will take the book to slide to the end of the table, t = 3.52 s

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